Backer board add not strength what so ever to the floor---use the 1/4" unless you want the extra height for some reason----
I nail floors with galvanized roofing nails---Rock screws work well but I see no added benefit on a floor---
Use a good quality modified thinset---color is up to you---
Yes to the 1/2" trowel and back buttering for the tile---
Consider renting a bridge saw and but a 4" angle grinder with a diamond blade---those are some long cuts--a smaller saw will work ,however--so use what you own.
I nail floors with galvanized roofing nails---Rock screws work well but I see no added benefit on a floor---
Use a good quality modified thinset---color is up to you---
Yes to the 1/2" trowel and back buttering for the tile---
Consider renting a bridge saw and but a 4" angle grinder with a diamond blade---those are some long cuts--a smaller saw will work ,however--so use what you own.